[FOM] formalism freeness

[Kennedy, Juliette]

Hi,


Thanks for the below invitation to discuss the ideas of my 2013 BSL paper, formalism freeness and entanglement in particular. (I have seen just a part of the thread so I am not sure who asked the question.)


>> > Related work by

>> > Kennedy (2013) suggests a pluralistic approach involving generalised
>> > constructibility and more widely the concept of "formalism freeness",
>> > and its dual, the concept of the entanglement of a semantically given
>> > object with its underlying formalism.
>>
>> I am a little bit familiar with generalized constructibility, and I
>> assume that "formalism freeness" means a deliberate non commitment to
>> any fixed formalism? I would like an elaboration of what
>> "entanglement" means here, e.g., by examples.
>>
>>
> I might invite Professor Kennedy to talk about this.
>


One of the things formalism freeness refers to is the stability of a semantically given structure under change of the underlying formalism.


The concept comes from Gödel's 1946 Princeton Bicentennial Lecture, in which he notes that with the concept of computability, many different formalisms define the same class of functions. He then asks whether the "formalism independence" (as he called it) exhibited by the concept of computability, might be obtained in the cases of provability and definability.


The concept of generalized computability was suggested in my 2013 paper as a possible implementation of this idea for the case of definability. With Magidor and Väänänen we were able to show that many different logics can be substituted for first order logic in the construction of L (and the same is true of substituting other logics for second order logic in the Myhill-Scott characterization of HOD). That is, L does not depend in an essential way on being defined in first order logic (and the respective is true of HOD).


On the other hand by substituting other logics for first order logic in the construction of L, one sees some interesting new intermediate models. See:


https://www.newton.ac.uk/files/preprints/ni16006_0.pdf


The logics we consider are fragments of second order logic, so this is also an investigation into the area that lies between first and second order logic. It is sometimes surprising that the logics that L "thinks" are first order, do not come close to exhibiting the usual Lindstrom properties. And conversely, logics which are very far from being first order, if measured by the Lindstrom properties, L "thinks" they are first order.


I used the term formalism freeness rather than formalism independence in order to refer to something much more general than what Gödel may have meant in his 1946 lecture.


Formalism freeness is a matter of degree, it is not a 0-1 concept like truth or proof.


The dual notion, "entanglement" has to do with something logicians are very familiar with, namely that a small change in the signature induces a massive change. For example, one has a 0-1 law for (finite) relational structures, but the 0-1 law fails if one allows function symbols in the language.


The intuition here is that in mathematical practice, one doesn't have this kind of entanglement with language, generally speaking.


All the best,

Juliette


Department of Mathematics and Statistics
P.O. Box 68 (Gustaf Hällströmin katu 2b)
FI-00014 University of Helsinki, Finland
tel. (+358-9)-191-51446, fax (+358-9)-191-51400
mobile: +358-50-371-4576

www.math.helsinki.fi/logic/people/juliette.kennedy/<http://www.math.helsinki.fi/logic/people/juliette.kennedy/>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20160424/f196962d/attachment.html>